THERY OF QUADRATIC EQUATIONS
SIP-1
1) Find the roots of the equation x²+ 9x -10= 0. -10,1
2) Find the roots of the equation 4x² -17x +4 = 0. 1/4,4
3) Find the nature of the roots of the equation 9x² -3x +1= 0. Complex
4) Find the nature of the roots of the equation 5x² - x -4= 0. Rational and unequal
5) If the sum of the roots of the equation kx² - 52x +24= 0 is 13/6, find the product of its roots. 24
6) If the roots of the equation 6x² - 7x + b= 0 are reciprocals of each other, find b. 6
7) The roots of a quadratic equation are a and - a. The product of its roots is -9. Form the equation in variable x. x²-9=0
8) The roots of the equation x² - 12x + k = 0 are in the ratio 1:2. Find k. 32
9) A quadratic equation has rational coefficients. One of the roots is 2+√2. Find its other root. 2-√2
10) I can buy 9 books less for Rs 1050 if the price of each book goes up by Rs 15. Find the original price and the number of books I could buy at that price. 35
11) P and Q are the roots of the equation x² - 22x +120 = 0. Find the value of
a) P²+ Q². 244
b) 1/P + 1/Q. 11/60
c) difference of P and Q. 2
12) If √(x +9) + √(x + 29)= 10. Find x. 7
13) 4ˣ⁺² + 4²ˣ⁺¹= 1280, find x. 2
14) The minimum value of 2x² + bx + c is known to be 15/2 and occurs at x= -5/2. Find the value of b and c. 10, 20
15) Find the number of positive and negative roots of the equation x² - ax + b = 0 where a> 0 , b> 0. 1 negative roots, 2 or 0 positive roots
16) If -1 and 2 are two of the roots of the equation x⁴+ 3x³+ 2x² +2x -4= 0. Then find the other two roots.
SIP-2
1) Find a quadratic equation whose roots are 3,4.
a) x² +7x +12= 0.
b) x² -7x +12= 0.
c) x² +7x -12= 0.
d) x² +7x -12= 0.
2) Find the roots of the equation x² -12x +13= 0.
a) 1,13 b) -1,-13 c) 6+√23, 6-√23 d) none
3) If the sum of the roots and the product of the roots of a equation are 13 and 30 respectively, find its roots.
1) 10,3 b) -10,-3 c) 10,-3 d) -10,3
4) Find the value of the discriminant of the equation 3x² +7x +2= 0.
a) 6.25 b) 25 c) 43 d) 5
5) Find the nature of the roots of the equation 2x² +6x -5= 0.
a) complex conjugate
b) real and equal
c) conjugate surds
d) unequal and rational.
6) Find the degree of the equation (x³- 3)² -6x⁵= 0.
a) 5 b) 6 c) 9 d) none
7) How many many roots (both real and complex) does (xⁿ - a)²= 0 have?
a) 2 b) n+1 c) 2n d) n
8) Find the signs of the roots of the equation x² +x -420= 0.
a) both are positive
b) both are negative
c) the roots are of opposite signs with the numerically larger root being positive
d) The roots are of opposite signs with the numerically root being negative.
9) Construct a quadratic equation whose roots are 2 more than the roots of the equation x² +9x + 10= 0.
a) x² +5x -4 = 0.
b) x² +13x +32= 0.
c) x² - 5x -4 = 0.
b) x² -13x +32= 0
10) Construct a quadratic equation whose roots are reciprocal of the roots of the equation 2x² +8x + 5= 0.
a) 5x² +8x +2= 0.
b) 8x² +5x +2= 0
c) 2x² +5x +8 = 0.
b) 8x² +2x +5= 0
11) The square of the sum of the roots of a quadratic equation E is 8 times the product of its roots. Find the value of the square of the sum of the roots divided by the product of the roots of the equation whose roots are reciprocals of these of E.
a) 8 b) 1/8 c) 1 d) 4
12) Construct a quadratic equation whose roots are one third of the roots of the equation x² +6x + 10= 0.
a) x² + 18x +90= 0.
b) x² +16x +80= 0
c) 9x² +18x +10= 0.
b) x² +17x +90= 0
13) Find the maximum value of the equation -3x² +4x +5.
a) 19/3 b) 31/12 c) 3/19 d) -19/3
14) The Quadratic expression ax² +bx +c has its maximum/ minimum value at
a) -B/2a b) b/2a c) -2b/a d) 2b/a
15) The expression (4ac - b²)/4a represents the maximum/minimum value of the expression ax²+ bx + c. Which of the following is true?
a) it represents the maximum value when a > 0.
b) it represents the minimum value when a < 0.
c) both a and b
d) neither a or b
16)
PERMUTATION
EX-1
ROAMING. 840
2) How many three letter words can be formed using the word PRACTICES. 357
3) In a party, each person shook hands with every other person present. The total number of hand shakes was 28. Find the number of people present in the party. 8
4) The letters of NESTLE are permuted in all possible ways.
a) How many of these words begin with T? 60
b) How many of these words begin and end with E ? 24
c) How many of these begins with S and end with L? 12
d) How many of these words neither begin with S nor end with L ? 252
e) How many of these words begin with T and do not end with N ? 48
5) The letters of word FAMINE are permuted in all possible ways.
a) How many of these words have all the vowels occupying odd places ? 36
b) How many of these words have all the vowels together? 144
c) How many of these words have atleast two of the vowels separated? 576
d) How many of these words have no two vowels next to each other? 144
6) Raju wrote 7 letters A,B,C,D,E,F and G on a black board.
a) How many 4 letters words can be formed using these letters such that atleast one letter of the word is a vowel? 720
b) How many 7 letters words can be made using these letters such that the letters at the words end are adjacent consonants? 720
c) How many 7 letters words can be formed using these letters such that the letter at one of its ends is a vowel and that at the other end is a consonant? 2400
7) All possible four digit number are formed using the digits 1,2,3 and 4 without repetition.
a) How many of these numbers have the even digits in even places? 4
b) If all the numbers are arranged in an ascending order of magnitude, find the position of the number 3241? 16
8) A committee of 5 is to formed from 4 women and 6 men.
a) In how many ways can it be formed if it consists of exactly 2 women? 120
b) In how many ways can it be formed if it consists of more women than men? 66
9) Find the number of four digits numbers which can be formed using four of the digits 0,1,2,3 and 4 without repetition. 96
10) The number of diagonals of a regular polygon is four times the number of its sides. How many sides does it have? 11
EX-2
1) The value of ⁸P₂ is
a) 28 b) 56 c) 48 d) 36
2) The value of ¹⁰C₂ is
a) 90 b) 20 c) 45 d) 50
3) The value of ⁴⁵C₄₂ is
a) ⁴⁵C₁ b) ⁴⁵C₄₁ c) ⁴⁵C₂ d) ⁴⁵C₃
4) The value of ²⁰⁰⁹C₀ is
a) 0 b) 1 c) 2009 d) 2008
5) The value of ²⁰⁰⁹C₁ is
a) 0 b) 1 c) 2009 d) 2008
6) The value of ²⁰⁰⁹C₂₀₀₈ is
a) 2009 b) 2008 c) 1 d) 0
7) If ⁿC₂ = ⁿC₁₀, then the value of n is
a) 10 b) 2 c) 8 d) 12
8) ⁸C₃ + ⁸C₄ = ⁿC₄, then n=
a) 4 b) 7 c) 9 d) 11
9) The relation between ⁿPᵣ and ⁿCᵣ is
a) ⁿPᵣ = ⁿCᵣ b) r. ⁿPᵣ = ⁿCᵣ c) ⁿPᵣ = r! ⁿCᵣ d) ⁿPᵣ . r! = ⁿCᵣ
10) The number of ways of arranging 6 people in a row is
a) 6 b) 30 c) 120 d) 720
11) The number of ways of arranging 10 books on a shelf such that two particular books are always together is.
a) 9!2! b) 9! c) 10! d) 8
12) The number of 3 digit numbers that can be formed using the digits 1,2,3,4,5,6 such that each digit occurs atmost once in every number is
a) 100 b) 60 c) 120 d) 20
13) Find the number of four digits numbers that can be formed using the digits 1,2,4,4,5,6 when each digit can occurs any number of times in each number.
a) 4⁶ b) ⁶P₄ c) ⁶P₅ d) 6⁴
14) Find the number of ways of posting 4 letters in 5 letter boxes.
a) 5⁴ b) 4⁵ c) 2⁵ d) 5²
15) Find the number of even numbers formed using all the digits 1,2,3,4,5 when each digit occurs only once in each number
a) ⁵P₄ b) 4!2 c) 5! d) 4!
16) Find the number of passwords of length 5 that can be formed using all the vowels of the alphabet
a) 120 b) 5 c) 3125 d) 25
* All the letters of the word RAINBOW are arranged in all possible ways.
17) Find the number of 7 letters words possible such that each letter is used atmost once.
a) 1 b) 24 c) 120 d) 7!
18) The number of 7 letters words that begin with R when each letter occurs only once is
a) 6,6! b) 7!2! c) 6! d) 2.7!
19) If each letter is used exactly once, the number of 7 letters words which begin with R and end with W is
a) 6! b) 5! c) 5!2! d) 4!
* Find the number of ways of arranging 6 people around it circular table
20) 6! 6!/2! c) 5! d) 5!/2!
21) Find the number of ways of selecting a team of 5 people from a group of 8.
a) ⁸C₃ b) ⁸P₅ c) 8! d) 5!
22) Find the number of ways of selecting a team of 6 people from a group of 10 people such that a particular person is always included in the team
a) ⁹C₅ b) ⁹C₆ c) ¹⁰C₅ d) ¹⁰C₆
23) Find the number of ways of selecting a team of 4 people from a group of 7 such that a particular person is not included in the team.
a) ⁷C₄ b) ⁶P₄ c) ⁶C₄ d) ⁶C₃
24) Find the number of ways of studding 10 beads to form a necklace
a) 9!/2! b) 9! c) 10!/2
25) Find the number of ways of inviting atleast one among 6 people to a party
a) 2⁶ b) 2⁶ -1 c) 6² d) 6² -1
26) The number of distinct lines that can be formed by joining 20 points on a plane is of which no three points are collinear is
a) 190 b) 380 c) 360 d) 120
27) Find the number of triangles that can be formed by joining 24 points on a plane, no three of which points are collinear?
a) 2024 b) 2026 c) 2023 d) 2025
28) The number of rectangles that can be formed on 8 x 8 chessboard is
a) 2194 b) 1284 c) 1196 d) 1296
29) The number of squares that can be formed on a 8 x 8 chessboard is
a) 204 b) 220 c) 240 d) 210
30) The number of ways of forming a committee of six members from a group of 4 men and 6 women is
a) 200 b) 210 c) 310 d) 220
1b 2c 3d 4b 5c 6a 7d 8c 9c 10d 11a 12c 13d 14a 15b 16c 17d 18c 19b 20c 21a 22a 23c 24a 25b 26a 27a 28d 29a 30b
EX-3
1) How many words can be found using all the letters of the word QUESTION without repetition so that the vowels are occupy the even places ?
a) 576 b) 720 c) 840 d) 1024 e) 620
2) In how many ways can the letters of the word RESULT be arranged so that the vowels appear in the even places only ?
a) 0 b) 48 c) 120 d) 144 e) 130
3) In how many ways can the letters of the world HEPTAGON be permuted so that the vowels are never separated
a) 720 b) 1440 c) 4230 d) 5040 e) 4320
4) Find the number of a ways in which the letters of the word INCLUDE can be permuted so that no two vowels appear together.
a) 7! - 5!3! b) 7! - 4!2! c) 4! 3 ! d) 4!5!/2 e) 4! 5!
5) which regular polygon has the ratio of its diagonals to its side as 3:1?
a) hexagon b) heptagon c) octagon d) nonagon e) pentagon
6) In how many ways can 5 prizes be given away to 3 boys when each boy is eligible for one or more prizes ?
a) 5³ b) è⁵ c) ⁵P₃ d) ⁵C₃ e) 242
7) in how many ways can one or more of 5 letters be posted into 4 mailboxes, if any letter can be posted into any of the boxes ?
a) 5⁴ b) 4⁵ c) 5⁵-1 d) 4⁵-1 e) 2⁸-1
8) How many four digit numbers having distinct digits can be formed using the digit 0 to 9 ?
a) 5040 b) 2526 c) 3656 d) 4365 e) 4536
9) in the above problem, how many of the numbers are divisible by 5 ?
a) 342 b) 504 c) 448 d) 952 e) 925
10) How many even number between 20000 and 40000 (excluding the extremes) can be found using the digit 0, 2, 3, 4, 6, 8 if any digit can occur any number of times ?
a) 2160 b) 2593 c) 2161 d) 2159 e) 2951
11) In how many ways can the letters of the world COMBINATION be permuted .
a) 11! b) 11!/(2!2!2!) c) 11!.(5!6!) e) 11!(2!2!2!5!) e) 11!/(2!2!)
12) How many four digit numbers that are divisible by 3 can be found using the digits 0, 2, 3, 5, 8 if no digit occurs more than once each number ?
a) 18 b) 22 c) 42 d) 66 e) 81
13) A certain group of friends a new year eve party and each person shook hands with everybody else in the group exactly once and the number of handshakes turned out to be 66. On the occasion of Pongal are (harvest festival). If each person in this group sends a greeting card to every other person in the group, then how many cards are exchanged
a) 33 b) 66 c) 132 d) 264 e) 123
14) If Mr Kapil one of the members of the group referred to in the previous question, wants to invite home one or more of his friends( from that group) for dinner. then in how many ways can invitation be extended ?
a) 1024 b) 2048 c) 4096 d) 2047
15) Find the number of selections that can be made by taking 4 letters from the word INKLING .
a) 48 b) 38 c) 18 d) 58
16) for the word discussed in the previous question, find the number of arrangements by taking 4 letters.
a) 270 b) 340 c) 460 d) 580 e) 480
17) A group of people is such that the number of ways of selecting 8 people is same as a number of a ways of selecting people . In how many ways can 18 people be selected from this group ?
a) 320 b) 240 c) 190 d) 80 e) 160
18)Manav Seva, a voluntary organisation has 50 members who plans to visit 3 slums in an area. They decide to divide themselves into three groups of 25, 15, and 10. In how many ways can the group division be made ?
a) 25!15!10! b) 50!/(25!15!10!) c) 50! d) 25!÷15!÷10! e) 50!/25!
* In how many ways can 20 different books be divided equally .
19) among 4 boys
a) 4⁵ b) 5⁴ c) 20!/(4!)⁴ d) 20!/(5!)⁴ e) 20!/(4!)⁴
20) into 4 parcels?
a) 20!/(5!(4!)⁴) b) 20!/(4!(5!)⁴) c) 20!/(5!. 4!)⁴ d) 20!/(5! . 4!) e) 20!/(5!)⁴
21) A boat is to be made by manned by eight men, of whom, one can not row on the bow side and two added cannot row on the stroke side. In how many ways can the crew be arranged ?
a) 2880 b) 1440 c) 4320 d) 5670 e) 5760
22) A double decker bus can accommodate 100 passengers , 60 in the lower deck and 40 in the the upper deck . In how many ways can 100 passengers be accommodated, if 15 of them want to be in lower deck only and 10 wants to be the upper deck only ?
a) (75!60!40!)/(45!30!)
b) (75!)/(45!30!)
c) (100!)/(60!40!)/0
d) (75!60!40!)/(25!50!) e) (75!/30!)
23) in how many ways can 12 differently coloured beads be to strung on a necklace ?
a) 12!12!/2!11!13!/211!/2
24) Sheetal inviting 10 of her friends for lunch and the places 5 of them at a round table and the remaining 5 at another round table. Find the total number of a ways in which she can arrange all are 10 friends .
a) (4!)² b) 10!(4!)²/(5!)² c) 10!(5!)² d) 10! e) 10!(5!)²/(4!)²
25) In how many ways can 6 boys and 6 girls sit around a circular table so that no two boys sit next to each other
a) (5!)² b) (6!)² c) 5!6! d) 11! e) 12!
26) If the letters of the word NOTES are permuted in all possible ways and the words thus obtained are arranged alphabetically as in dictionary, then what is the rank of the word STONE?
A) 95 b) 96 c) 105 d) 106 e) 94
27) Find the sum of all four digit numbers formed by taking all the digits 2,4,6,8
a) 133320 b) 533280 c) 244420 d) 335240 e) 132320
28) There are 12 points on a plane. If 4 of them are on in a straight line and no other three points are on a straight line, then find the difference between the number of triangles and the number of straight lines that can be formed these points.
a) 215 b) 216 c) 156 d) 156 e) 515
29) In how many ways can a panel of 6 doctors be formed from 5 surgeons and 6 physicians if the panel has to include more surgeons then physicians?
a) 82 b) 81 c) 65 d) 135 e) 89
30) in how many ways can a delegation of 4 professors and Three students be constituted from 8 professors and 5 students. If Balamurthy an arts students refuses to be in the delegation when Prof. Siddharth, the science professor is included in it?
a) 280 b) 210 c) 490 d) 560 e) 620
31) Neha has 12 chocolates with her; four similar kit kar, five similar Perks and three similar milk bars, which she wants to distribute among her friends. In the how many ways can Neha give away one or more chocolates ?
a) 120 b) 119 c) 60 d) 59 d) 130
32) Ram attempts a question paper that has three sections with 6 questions in each section. If Ram has to attend any 8 questions, chosing at least two questions from each section, then in how many ways can he attempt the paper
a) 18000 b) 10125 c) 28125 d) 9375 e) 28521
33) A password of length 5 is to be formed using one or more of the symbols {a,b,c,d, @, #, 1, 2, 3}. How many of these follow a palindrome pattern ? (palindrome is a word that reads the same backward or forward )
a) 147 b) 6561 c) 5184 d) 749 e) 59949
34) Find the number of non negative integers which satisfies the equation x₁ + x₂ + x₃ + x₄ = 15.
a) 216 b) 165 c) 364 d) 316 e) 816
35) In how many ways can the letters of the word SUBJECT be placed in squares of the given below so that no row remains empty ?
a) 5x 6! b) 10! x 6! c) 11x 5! d) 13 x 8! e) 10x 8 !
36) The number of positive integers of solutions to the equation x+ y+ z = 20 is
a) 131 b) 110 c) 55 d) 171 e) 141
1a 2d 3e 4d 5d 6b 7c 8e 9d 10d 11b 12c 13c 14e 15d 16a 17c 18b 19d 20b 21e 22b 23e 24b 25c 26b 27a 28d 29b 30c 31b 32c 33d 34e 35d 36d
Exercise - 4
1) Consider the set A={a,b,c,d,e,f,g,h}. Find the number of subset of A which contains at least six elements and including c and e
a) 20 b) 21 c) 22 d) 64 e) 256
2) Find the number of ways of arranging the letters of the word CALENDAR in such a way that exactly two letters are present in between L and D ?
a) 2640 b) 3000 c) 2600 d) 7200 e) 3600
3) In how many ways, can the letters of the word EUROPE be arranged so that no two vowels are together?
a) 12 b) 24 c) 360 d) 300 e) none
4) Raju has forgotten his 6 digit ID number. He remember the following the first two digits are either 1, 5 or 2, 6, the number is even and 6 appears twice . if Raju uses a trial and error process to find his ID number at the most, how many trials does he need to succeed ?
a) 972 b) 2052 c) 729 d) 2051 e) 243
5) A four digit number using the digits 0, 2, 4, 6 8 without repeating any one of them. What is the sum of all possible numbers ?
a) 519960 b) 402096 c) 13320 d) 4321302 e) 5333280
6) How many four digit odd numbers can be formed , such that every 3 in the number is followed by 6 ?
a) 108 b) 2592 c) 2696 d) 2700 e) 100
7) How many four digit numbers are there between 3200 and 7300 in which 6,8 and together or separately do not appear ?
a) 1421 b) 1420 c) 1422 c) 3600 d) 1077
8) How many time does the digit 5 appear in the numbers from 9 to 1000 ?
a) 300 b) 257 c) 256 d) 243 e) 299
9) A matrix with four rows and three columns is to be formed with the entries 0, 1, 2. How many such district metrices are possible.
a) 12 b) 36 c) 3¹² d) 2¹² e) 3¹²-1
10) There are 5 bowls numbered 1 to 5, 5 green balls in 6 black balls. Each of bowl is to be filled by either a green or a black ball and no two adjacent bowls can be filled by green balls. if the same colour balls are indistinguishable, then the number of different possible arrangements is
a) 8 b) 7 c) 13 d) 256 e) 15
11) How many four digit numbers can be formed such that the digit in the 100th place is greatest than that in the 10th place ?
a) 9000 b) 10000 c) 4500 d) 4050 e) 2250
12) In how many ways can 4 post cards be dropeed into to 8 letter boxes ?
a) ⁸C₄ b) 4⁸ c) 8⁴ d) 24 e) none
13) The number of positive integeral solutions of the equation a+ b + c + d= 20 is
a) 1771 b) 1331 c) 256 d) 512 e) 969
14) There are 4 identical oranges, 3 identical mangoes and two identical apples in the basket. The number of ways in which we can select one or more fruits from the basket is
a) 60 b) 59 c) 57 d) 55 e) 56
15) In how many ways can 5 boys 3 girls sit around a table in such a way that no 2 girls sit together ?
a) 480 b) 960 c) 320 d) 1500 e) 1440
16) Find the maximum number of a ways in which the letters of the word MATHEMATICS can be arranged so that all Ms are together and all Ts are together.
a) 11! b) 11!/(2!2!2!) c) 9/(2!2!2!) d) 7!2! e) 5!/2!
17) In how many arrangements of the word MATHEMATICS, the two A's are separated ?
a) 10!/(2!2!2!) b) 9!/(2!2!2!) c) 9 x 10! d) 111!/(2!2!2!) e) (9 x10!)/(2!2!2!)
* Considere the words INSTITUTE
18)A in how many ways can 5 letters. be selected from the word ?
a) 41 b) 33 c) 36 d) 40 e) 55
B) In how many arrangements can be made by taking 5 letters from the word ?
a) 2790 b) 8730 c) 4320 d) 7200 e) 2250
19) The letters of the word AGAIN are permuted in all possible ways and are arranged in dictionary order. What is the 28th word ?
a) GAIAN b) GAINA c) GANIA d) NGAIA e) AGANI
20) If all possible 5 digit numbers that can be formed using the digit code 4, 3, 86 and 9 without repetition are arranged in the ascending order, then the position of the number 89634 is
a) 91 b) 93 c) 95 d) 98 e) 100
21) in which regular polygon, is the number of diagonals equals to two and half times the number of sides ?
a) heptagon b) Pentagon c) decagon d) octagon e) none
22) in how many ways can 12 distinct pens be divided equally
A) among 3 childrens ?
a) 12/(3!)⁴ b) 12/((4!)⁴ 3!) c) 12!/3!4! d) 12!/(4!)³. e) 12!/(4!)⁴
B) into 3 parcels ?
a) 12/(4!)⁴ b) 12/(4!)³ c) 12!/(3!4!). d) 12!/(4!)³3! e) 12/(3!)⁴
23) In a certain question paper, a candidate is required to answer 5 out of 8 questions, which are divide into two parts containing 4 questions each. he is permitted to attempt not more than three from any group. The number of ways in ways he can answer the paper is
a) 24 BC) 96 c) 48 d) 84 e) 32
24) The sides PQ, QR and RS of ∆ PQR have 4, 5 and 6 points (not the end points) respectively on them. The number of triangles that can be constructed using these points as vertices is
a) 455 b) 34 c) 425 d) 65 e) 421
25) There are eight different books and two identical copies of each in a library. The number of ways in which one or more books can be selected is
a) 2⁸ b) 3⁸-1 c) 2⁸-1 d) 3⁸ e) none
26) The number of 4 digit telephone numbers that have at least one of their digits repeated is
a) 9000 b) 4464 c) 4000 d) 3986 e) 4536
27) we are given three different green dyes, 4 different red dies and two different yellow dies. The number of ways in which the dice can be chosen so that at least one green dye and one yellow die is selected is
a) 336 b) 335 c) 60 d) 59 e) none
28) There are 5 balls of different colours and 5 boxes of colours the same as those the balls. The number of ways in which the balls, one in each box can be placed such that a ball does not go to a box of its own colour is
a) 40 b) 44 c) 42 d) 36 e) 34
29) P is a integer whose digits are zeros and ones. The sum of the digit of P is 4 and 10⁵< P < 10⁶.
How many values P can take ?
a) 79 b) 60 c) 10 d) 20 e) 30
30) A question paper contains of 5 problems, each problem having three internal choices. In how many ways can a candidate attempt one or more problems ?
a) 63 b) 511 c) 1023 d) 15 e) 31
31) 6 points are marked on a straight line and 5 points marked on another line which is parallel to the first line . How many straight lines, including the first two , can be formed with these points
a) 29 b) 33 c) 55 d) 30 e) 32
32) The number of sequences in which 7 players can throw a ball, so that the youngest player may not be the last is
a) 4000 b) 2160 c) 4320 d) 5300 e) 4160
33) Sixteen guests have to be seated around two circular tables each accommodating 8 members, 3 particular guests desire to sit at one particular table and 4 others at the other table. The number of ways of arranging these guest is
a) ⁹C₅ b) (9! x 7!)/(4!5!) c) 9!(7!)²/(4! 5!) d) (7!)² e) none
34) In how many ways is it possible to choose two white squares so that they lie in the same row or same column on an 8x8 chess board?
a) 12 b) 48 c) 96 d) 60 e) 100
35) The number of non negative integral solutions to the equation a+ b + c= 14 is
a) 78 BC) 45 c) 120 d) 110 e) 126
1c 2e 3e 4b 5a 6c 7e 8e 9c 10c 11d 12c 13e 14b 15e 16e 17e 18A) a B) b 19b 20c 21d 22a) d b) d 23c 24e 25b 26b 27a 28b 29e 30c 31e 32c 33c 34c 35c
LOGARITHM
EXERICISE- A
1) Simplify:
a) log315+ 4 log25 - 6 log9 - 3 log 49.
b) log700+ log1280+ 3 log25.
2) Solve for x:
a) log₁₀20x = 4. 500
b) log3x - log6= log12. 24
c) log(x +3)+ log(x -3)= log72. 9
3) Express log√a³/(b⁶c⁴) in terms of loga, log b and logc.
4) From the number of digits in 294²⁰ given that log6= 0.778 and log7 = 0.845. 50 digits
5) Obtain an equation between x and y, without involving logarithms, if 3 log x = 4 log y +5. x³= 10⅝y⁴
6) Find the value of log₃√₂ 32 ³√16. 19
7) Find the number of zeros after the decimal point in (3/4)⁵⁰⁰, given that log 3 = 0.4771 and log 2= 0.3010. 62
8) If log2= 0.301, find the value of log1250, log 0.001250 and log 12500.
Exercise - B
1) Simplify log(₃₂)(₁₈)(48)(12).
a) 1 b) 2 c) 1/2 d) none
2) If logᵥa = log꜀a where a is natural number and both v and c exceeds a, which of the following, is true?
a) b is equal to c
b) b is not equal to c.
c) b need not be equal to c
3) log₃4+ log₃16= log₃x. Find x.
a) 64 b) 4 c) 12 d) 20
4) Log₂72 - log₂3 = log₂x. Find x.
a) 69 b) 75 c) 24 d) 216
5) What is the value of log₃x⁰ where x≠ 0.
a) 1 b) 0 c) 3 d) -1
6) log₉27²=
a) 3 b) 6 c) 1 d) 2
7) If log₂₇8(logₓ3)= 1, find x
a) 2 bb) 4 c) 8 d) 16
8) (log₁₁64)/(log₁₁81)=
a) log₃2 b) log₂3 c) (3 log2)/log3 d) log₉8
9) If 4 log₄5²= x, find x.
a) 25 b) 5 c) 1 d) 16
10) Find the integral part of log₂20000.
a) 4 b) 5 c) 14 d) 15
11) If N is a 18 digit number, find the integral part of log₁₀N.
a) 17 b) 18 c) 19 d) none
12) What is the value of log₁/₅ 0.0000128?
a) -7 b) -5 c) 5 d) 7
13) If x is the product of the logarithms of the first 10 natural numbers, which of the following is true?
a) x=1 b) x >1 c) x <1
14) If a> 1, logₐa + log√ₐ a + log³√ₐ a+....+ log²⁰√ₐ a =
a) 420 b) 210 c) 380 d) 190
15) log₀.₀₆₂₅2=
a) -1/4 b) 1/4 c) 4 d) -4
Exercise - C
₇₇⁵ₐˣᵥˣₐᵛ²₁₀ₑₘʸᵐₑᵥᵥᵥᵥᵥ∞ ∞∞₁₀₁₀₂₀₁₀₁₀₁₀₂₂ₓ¹²ₐᵥ꜀ₓᵧₖₓ₅ˣʸᶻ²ᵥ꜀ᵧₖᶻˣ₂²ˣ⁻²ˣₑ⁶⁶ᶻ⁶ᵥ꜀ₐ꜀ₐᵥ¹²₁₂₆⁴³⁴³²³³¹⁾⁶ₓʸᶻ₆₆₇₄₃₃⁶³²⁴ˣₓₓ₁₆ₓ₆₄₂ˣ₄¹⁻ˣ₂₂₂₂ₐₐ₃₂₀¹⁾⁴⁰⁰ₓ²¹⁰²⁰ₑₐ⁴⁴ₓ₄₉⁵₁₀⁵¹⁰₁₀₁₀⁸⁵⁰¹⁻ˣ⁴⁰⁰³³²²³³²²
₂₅₁₂₅₂₅₀₂₀₀₀₈³¹⁾³ʸˣₓ₊ᵧᵣᵣ²²₂₁₀₁₀₂₀ₓₓ₂ₓ₃ᵧᵥ꜀ₐ꜀ₐᵥₐ₇₇₀₁₂₅³³⁹⁹³₄₄ᵐ₄ᵐₓₓₓₖₓₖₓʸᶻˣₖₓᵧ²³⁴⁵₃²₃ₓₓₓₓ₅ₓ₄₉ₓₓₘₙₘₙₚₚₘₙₘ₊ₙ¹₁₀₁₀²⁵⁰
